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Error bounds for the long‐range attractive potential between pairs of atoms and molecules have been calculated by constructing oscillator strength distributions which reproduce known dynamic polarizabilities, oscillator strengths, and photoionization cross sections. Results from the theory of moments are applied to construct these distributions so that they provide rigorous error bounds on the force constant. Approximate solutions to the moment equations which result are found by means of linear programming. Results obtained for rare‐gas interactions are found to be consistent with known values, with error bounds significantly tighter than those obtained from previous calculations. These same oscillator strength distributions are used to calculate error bounds on the dynamic polarizabilities for frequencies between those of the existing measurements and the first dipole‐allowed transition.

In this paper we are going to study infinite systems of ordinary differential equations We are specifically interested in solving systems of equations that arise as moment equations associated with a Fokker-Planck [l] or pseudo Fokker- Planck [2; 3, Eq. (S)] e q ua t ions. These equations occur frequently in statis- tical physics. The moment problems we will consider are associated with a problem of the form (W4 (t) = A(t) u(t), 40) = *o 3 u(t) = 24(x, t) (1.2) A(t) = c a&) xp(d/dx)q.

Macroscopic models based on moment equations are developed to describe the transport of mass and energy near the phase boundary between a liquid and its rarefied vapour due to evaporation and hence, in this study, condensation. For evaporation from a spherical droplet, analytic solutions are obtained to the linearised equations from the Navier–Stokes–Fourier, regularised 13-moment and regularised 26-moment frameworks. Results are shown to approach computational solutions to the Boltzmann equation as the number of moments are increased, with good agreement for Knudsen number ${\lesssim}1$, whilst providing clear insight into non-equilibrium phenomena occurring adjacent to the interface.

A simple derivation and extension of the bounds on dynamic polarizabilities obtained by Langhoff, Karplus, and Gordon are given. A new matrix method to solve the relevant moment equations is presented.

A systematic method of obtaining closure relations for Maxwell's moment equations is developed, and shown, by application to the problem of an infinitely strong shock wave, to lead to results at the fourth order truncation level which agree with Mott–Smith and Monte Carlo predictions.

A technique is described, involving integration by parts, for constructing Liapunov functions for a class of third- and fourth-order differential equations. The procedure makes use of a number of "differential moments" M_{n} = \int\min{t_{0}}\max{t} (d^{n}x/dt^{n})L[x]dt , where L[x] = 0 is the differential equation under consideration. The moment equations M_{n}=0 (n=0, 1, 2, ...) are manipulated to obtain a single expression [V(x,\dot{x},\ddot{x},... )]\min{t_{0}}\max{t} = \int\min{t_{0}}\max{t} k(x,\dot{x},\ddot{x}...)dt . Conditions are applied to the functions involved in L[x] so as to make V positive definite and k negative semidefinite, in which case V(x,\dot{x},\ddot{x} ... ) will be a Liapunov function for L[x]=0 .

In the letter a new method for the resolution of the moment equations in a nonlinear optical amplifier is exposed. Results for the four first moments for the saturated and unsaturated absorption cases are presented. These results are in good agreement with those obtained by other authors.

An analysis of Thome’s (1981) structure equations governing radiation field moments is presented in the case of stationary, spherically symmetric flows. It is shown that the truncated system of moment equations must be solved subject to boundary conditions and a number of critical points equal to the truncation order l max arises. The moment formalism can be made tractable once a suitable closure assumption for moments of order higher than l max is introduced. A sample numerical integration of the moment equations is performed for a spherical, accreting flow with an assigned velocity and temperature profile.

Diplomarbeit, Rheinisch-Westfälische Technische Hochschule Aachen, 1989; Diplomarbeit, Rheinisch-Westfälische Technische Hochschule Aachen, 1989, The hot electron transport in the ballistic regime is calculated for inhomogeneously doped sub-micron structures [0.4 μm] of GaAS, based on the coupled Boltzmann transport - and Poisson equations. We use for the solution of this integrodifferential equation system the moment method, with a six equations hiearchy. Numerically, we apply for the first time the homotopy analysis method for the solution of this nonlinear equation system which allows us to describe the strong non-equilibrium electron transport in three dimensions. Thereby, for the first time, the three dimensional semi-classical electron transport can be calculated together with quantum-mechanical collision operators, here acoustic electron-phonon as well as electron-impurities interactions. With this three-dimensional electron transport Boltzmann‘ moment equations we are able to calculate the behaviour of temperature variation orthogonal to the direction of transport.

Chapter 1 considered the problem of finding the pressure p (x, y, z, t) in regions occupied by the fluid masses following the specified elastic displacements \(\overrightarrow{u}\) (x, y, z, t). Here, let us consider the problem of finding these displacements for the given pressure distribution p (x, y, z, t)in regions occupied by the fluid masses. According to D’Alembert’s principle, equations defining the vectorial function \(\overrightarrow{u}\)(x, y, z, t) can be set up as equilibrium equations for an elastic body by supplementing the volume forces acting on the body by the volume inertia forces. The volume force vector \(\overrightarrow{w}\)(x,y, z, t) in this case will have the form $$\mathop \omega \limits^ \to= \mathop \omega \limits^ \to+ \frac{{\mathop {d\omega }\limits^ \to}}{{dt}}x\mathop r\limits^ \to+ \frac{{{\partial ^2}u}}{{\partial {t^2}}}$$ (2.1.1) where \(\overrightarrow{w}\) (x,y, z, t) is the acceleration of a particle of the elastic body with coordinates x, y, z at time t. It is assumed that the moving coordinate system x,y, z rotates slowly. Disregarding the Coriolis acceleration and the acceleration of transport \(\overrightarrow{w}\times \left( \overrightarrow{w}\times \overrightarrow{r} \right)\), we will set in Eq. (2.1.1) $$\overrightarrow{w}={{\overrightarrow{w}}_{0}}+\frac{d\overrightarrow{w}}{dt}\times \overrightarrow{r}+\frac{{{\partial }^{2}}\overrightarrow{u}}{\partial {{t}^{2}}}$$ (2.1.2) in the same manner as in deriving the force and moment equations in Chap. 1. Equation (2.1.1) in this case will take the form $$\overrightarrow{Q}=-\rho \left( {{\overrightarrow{w}}_{0}}-\overrightarrow{g}+\frac{d\overrightarrow{w}}{dt}\times \overrightarrow{r}+\frac{{{\partial }^{2}}\overrightarrow{u}}{\partial {{t}^{2}}} \right)$$ (2.1.3)

In this paper we shall first outline what we consider to be the statistical solution to a problem in which the material medium is described from a stochastic point of view. Subsequent to this we shall discuss the hierarchy of moment equations that result from averaging the basic governing equation. Here we indicate the solution for small perturbations in the elastic constants when the material is statistically homogeneous and isotropic and the average stress field is uniform. Effective constants will next be defined. Using variational principles and the statistical perturbation solutions previously obtained, expressions for bounds will be determined which contain higher order statistical information. The bulk modulus will be considered as an example. Finally, simple algebraic formulas will be obtained for the bounds by using a cell model for materials. The cell model was applied by Miller.(1)

We study the Darcy boundary value problem with lognormal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the stochastic solution of the PDE by the expected value of its Taylor polynomial. The recursive deterministic equation satisfied by the expected value of the Taylor polynomial (first moment equation) is formally derived. Well-posedness and regularity results for the recursion are proved to hold in Sobolev space-valued Holder spaces with mixed regularity. The recursive first moment equation is then discretized by means of a sparse approximation technique, and the convergence rates are derived.

Switching criteria for hybrid hydrodynamic/molecular gas flow solvers are developed, and are demonstrated to be more appropriate than conventional criteria for identifying thermodynamic non-equilibrium. For switching from a molecular/kinetic solver to a hydrodynamic (continuum-fluid) solver, the criterion is based on the difference between the hydrodynamic near-equilibrium fluxes (i.e. the Navier–Stokes stress and Fourier heat flux) and the actual values of stress and heat flux as computed from the molecular solver. For switching from hydrodynamics to molecular/kinetic, a similar criterion is used but the values of stress and heat flux are approximated through higher order constitutive relations; in this case, we use the R13 equations. The efficacy of our proposed switching criteria is tested within an illustrative hybrid kinetic/Navier–Stokes solver. For the test cases investigated, the results from the hybrid procedure compare very well with the full kinetic solution, and are obtained at a fraction of the computational cost.

We derive a closed‐form expression for mean travel time of a conservative solute migrating under uniform in the mean flow conditions within an infinite stationary field with simple exponential correlation of the natural logarithm of hydraulic conductivity. Our expression is developed from a consistent second‐order expansion in σY(standard deviation of the log hydraulic conductivity) of the equations for moments of travel time and trajectories of conservative solutes in two‐dimensional randomly nonuniform flows ofGuadagnini et al.[2001]. As such, it is nominally valid for moderately heterogeneous fields, with σY2< 1. Its validity for larger heterogeneity degrees is tested against numerical Monte Carlo simulations. Our results clarify the nonlinear effect in the mean travel time with respect to distance that has been observed numerically (and modeled empirically) in the literature.

We study the Darcy problem with log-normal permeability, modeling the fluid flow in a heterogeneous porous medium. A perturbation approach is adopted, expanding the solution in Taylor series around the nominal value of the permeability. The resulting recursive deterministic problem satisfied by the expected value of the stochastic solution, analytically derived and studied in [4], is discretized with a full tensor product finite element technique. To overcome the incurred curse of dimensionality the solution is sought in a low-rank tensor format, the so called Tensor Train format. We develop an algorithm for solving the recursive first moment problem approximately in the Tensor Train format and show its effectiveness with numerical examples.

Analytical solutions of the Rankine–Hugoniot conditions for a radiating gas are presented both for shocks and subshocks. The case of Synge gas is studied in detail. In order to study subshocks, radiation is described by the moment equations closed with a variable Eddington factor.

We study the mixed formulation of the stochastic Hodge-Laplace problem defined on a n-dimensional domain D ($n \geq 1$), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three dimensional case. We derive and analyze the moment equations, that is the deterministic equations solved by the m-th moment ($m \geq 1$) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order of convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces.

In this note, we present the derivation of moment equations for a deterministic gene expression model with rational right-hand side, which is subject to initial condition and parameter uncertainty. Our derivation is based on the use moment-generating functions and the application of the transport theorem [1]. [1] M. Ehrendorfer,The Liouville equation and atmospheric predictability. Cambridge University Press, 2006, p. 59–98

In this paper the solution of the Fokker Planck (FPK) equation in terms of (complex) fractional moments is presented. It is shown that by using concepts coming from fractional calculus, complex Mellin transform and related ones the probability density function response of nonlinear systems may be written in discretized form in terms of complex fractional moment not requiring a closure scheme.

Eight of the Chernikov-Grad 13-moment equations, determining the approximate solution of the general relativistic Boltzmann equation have been developed completely including the nonstationary thermodynamical problem. Entropy balance and Gibbs equations were also formulated at this nonstationary level.

The velocity slip and temperature difference are examined using two‐temperature thirteen moment equations. It is shown that the velocity slip becomes maximum just beyond the throat and the temperature difference is appreciable in the supersonic flow regions.

The shape of the envelope soliton of random phase waves, which is considered to have an arbitrary form, is obtained using moment equations of the wave kinetic equation.

A study is made of the flow of heat between parallel plates of slightly different temperatures. The problem is described by the linearized Boltzmann equation which is subject to microscopic boundary conditions. We approximate the distribution function by half‐range polynomials in velocity space and determine the space‐dependent coefficients by forming half‐range moment equations. An approximation involving four pairs of space functions suffices to give an accurate treatment of the heat flow and of the density and temperature profiles for the entire range of conditions from free molecule to hydrodynamic. Detailed numerical results for the temperature slip and molecular boundary structure are obtained for hard‐sphere molecules. The accuracy of cruder half‐range approximations and other methods of fixing the coefficients is established.

We show that the partial differential equations obtained by Tatarski, Beran, and Ho for the first, second, and fourth statistical moments of a wave propagating in a random medium are special cases of a general partial differential equation satisfied by the mth moment when the refractive-index inhomogeneities are sufficiently weak. In addition, we derive moment equations that apply regardless of the strength of the inhomogeneities. The results obtained in this case are in the form of difference relations satisfied for the moments.

A mixture of two Maxwell molecule gases having different temperatures, but uniformly distributed in space is considered. The exact equations for the second‐ and fourth‐order moments are developed; no truncation is necessary for this problem. The temperatures of the two species are readily found. The fourth‐order moment equations are solved, with details given for the cases of identical molecules, and of large molecular mass ratio. The coefficients in generalized orthogonal polynomial expansions are then found in terms of the moments; the zero‐order distributions used are Maxwellian with the initial gas temperatures, the final temperature, or the time‐dependent temperature. The meaning for the various forms for the coefficients is discussed; it is shown that only in the case of large mass‐ratio gas mixtures does a Maxwell distribution always remain Maxwellian.